# Sum of minimum number of vertices

Geometry Level 5

At $$t = 0$$, I begin truncating a cube. At $$t = 1$$, I obtain its dual, the octahedron.

Find

$\sum_{S \in 2^{\lbrace 0,1 \rbrace}} \min_{t \in (0,1) \cup S} V(t)$

where $$V(t)$$ is the number of vertices of the truncated cube at time $$t$$ and $$2^{A}$$ is the power set of $$A$$.

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